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A z-test is used for testing the mean of a population versus a standard, or comparing the means of two populations, with large (n ≥ 30) samples whether you know the population standard deviation or not.
Z-tests can also be helpful when we want to test a hypothesis. Generally, they are most useful when the standard deviation is known.
T-tests are calculations used to test a hypothesis, but they are most useful when we need to determine if there is a statistically significant difference between two independent sample groups
T-tests are most appropriate when dealing with problems with a limited sample size (n < 30).
Both Z-tests and T-tests require data with a normal distribution, which means that the sample (or population) data is distributed evenly around the mean, just like in this figure: